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arXiv:2411.03640 · Science Advances

Efficient learning of mixed-state tomography for photonic quantum walk

Qin-Qin Wang, Shaojun Dong, Xiao-Wei Li, Xiao‐Ye Xu, Chao Wang, Shuai Han, Man‐Hong Yung, Yong‐Jian Han, Chuan‐Feng Li, Guang‐Can Guo

Published Mar 15, 2024• 11 citations
arXivHTMLPDF

Abstract

Noise-enhanced applications in open quantum walk (QW) has recently seen a surge due to their ability to improve performance. However, verifying the success of open QW is challenging, as mixed-state tomography is a resource-intensive process, and implementing all required measurements is almost impossible due to various physical constraints. To address this challenge, we present a neural-network-based method for reconstructing mixed states with a high fidelity (∼97.5%) while costing only 50% of the number of measurements typically required for open discrete-time QW in one dimension. Our method uses a neural density operator that models the system and environment, followed by a generalized natural gradient descent procedure that significantly speeds up the training process. Moreover, we introduce a compact interferometric measurement device, improving the scalability of our photonic QW setup that enables experimental learning of mixed states. Our results demonstrate that highly expressive neural networks can serve as powerful alternatives to traditional state tomography.

DOI
10.1126/sciadv.adl4871

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Quantum state tomography (QST) is a fundamental technique in quantum information science, used to characterize the density matrix of an unknown quantum state. However, traditional QST methods are often computationally intensive, requiring a large number of measurements and sophisticated data analysis techniques. This paper proposes a novel approach utilizing self-organizing neural networks (SONNs) for direct quantum state reconstruction. The core idea is to train a SONN to learn the mapping between experimental measurement data and the corresponding density matrix, thereby bypassing the need for precise control over the quantum system and significantly reducing the computational burden. We detail the architecture of the SONN, the training process, and demonstrate the potential of this method to achieve accurate state reconstruction with fewer measurements. The proposed method offers a promising path towards more efficient and scalable QST, particularly for complex quantum systems.
We apply deep-neural-network-based techniques to quantum state classification and reconstruction. Our methods demonstrate high classification accuracies and reconstruction fidelities, even in the presence of noise and with little data. Using optical quantum states as examples, we first demonstrate how convolutional neural networks (CNNs) can successfully classify several types of states distorted by, e.g., additive Gaussian noise or photon loss. We further show that a CNN trained on noisy inputs can learn to identify the most important regions in the data, which potentially can reduce the cost of tomography by guiding adaptive data collection. Secondly, we demonstrate reconstruction of quantum-state density matrices using neural networks that incorporate quantum-physics knowledge. The knowledge is implemented as custom neural-network layers that convert outputs from standard feed-forward neural networks to valid descriptions of quantum states. Any standard feed-forward neural-network architecture can be adapted for quantum state tomography (QST) with our method. We present further demonstrations of our proposed QST technique with conditional generative adversarial networks (QST-CGAN) [Ahmed et al., Phys. Rev. Lett. 127, 140502 (2021)]. We motivate our choice of a learnable loss function within an adversarial framework by demonstrating that the QST-CGAN outperforms, across a range of scenarios, generative networks trained with standard loss functions. For pure states with additive or convolutional Gaussian noise, the QST-CGAN is able to adapt to the noise and reconstruct the underlying state. The QST-CGAN reconstructs states using up to two orders of magnitude fewer iterative steps than iterative and accelerated projected-gradient-based maximum-likelihood estimation (MLE) methods. We also demonstrate that the QST-CGAN can reconstruct both pure and mixed states from two orders of magnitude fewer randomly chosen data points than these MLE methods. Our paper opens possibilities to use state-of-the-art deep-learning methods for quantum state classification and reconstruction under various types of noise.
Quantum state tomography is defined as a process of reconstructing the density matrix of a quantum state and is an important task for various emerging quantum technologies. In this work, we propose a general quantum state tomography framework that employs deep neural networks to reconstruct quantum states from a set of measurements with high efficiency. In particular, we apply it to two cases, including few measurement copies and incomplete measurement. Numerical results demonstrate that the proposed method exhibits a significant potential to achieve high fidelity for quantum state tomography when measurement resources are limited.
Abstract Quantum tomography is currently ubiquitous for testing any implementation of a quantum information processing device. Various sophisticated procedures for state and process reconstruction from measured data are well developed and benefit from precise knowledge of the model describing state-preparation-and-measurement (SPAM) apparatus. However, physical models suffer from intrinsic limitations as actual measurement operators and trial states cannot be known precisely. This scenario inevitably leads to SPAM errors degrading reconstruction performance. Here we develop a framework based on machine learning which generally applies to both the tomography and SPAM mitigation problem. We experimentally implement our method. We trained a supervised neural network to filter the experimental data and hence uncovered salient patterns that characterize the measurement probabilities for the original state and the ideal experimental apparatus free from SPAM errors. We compared the neural network state reconstruction protocol with a protocol treating SPAM errors by process tomography, as well as to an SPAM-agnostic protocol with idealized measurements. The average reconstruction fidelity is shown to be enhanced by 10% and 27%, respectively. The presented methods apply to the vast range of quantum experiments which rely on tomography.